We analyze peculiarities of the Pauli principle in thrre-cluster system. We demonstrated that the antisymmetrization operator, being a source of the three-body interaction, can be decomposed into components involving either permutations between all three clusters, or permutations between two clusters only. Introducing this into the Faddeev equations, one obtains three alternative but equivalent formulations. These sets of equations are presented in operator form, for practical applications we will make use full set of the oscillator functions.